Path Integrals Emergent from Observer-Equivariance and G-Symmetry

The idea for this article originates from an attempt to reconstruct the laws of physics from Leibniz’s ontology. In this ontology, the observer can be understood as a windowless monad. Leibniz’s ontology also includes the concept of the “pre-established harmony.” I realized that the symmetry principles in physics, within a Leibnizian ontology, can play a role analogous to the pre-established harmony. I also recognized that the relationship between the observer and the physical system can be expressed as a G-principal bundle p: O -> S, where O is the set of all observer perspectives and S is the physical system. For observers to share a coherent reality, the morphisms between any two observer perspectives must be invertible. In category-theoretic language, this implies isomorphism and G-equivariance. This article is largely generated by AI based on these insights. My first draft of this article was created with ChatGPT version o3 using the prompt “Derive Feynman’s path integrals from G-equivariance.” This is the first working draft of the manuscript. Although it already stands as a complete draft, the true process lies in making it more human by integrating the content into my own understanding. Each revision involves internalizing the concepts, refining the narrative, and ensuring that the final text genuinely reflects my insights rather than merely an AI-generated framework. Through successive iterations, this draft will evolve into a seamless expression of my original ideas.